Infinitely Divisible Distributions with Unimodal Levy Spectral Functions

Abstract

The class of infinitely divisible characteristic functions which have unimodal Levy spectral functions is determined. It is shown that membership in this class is related to solutions of the equations $\phi(u) = \phi^r(ru)\phi_r(u)$, where $r \in (0, 1)$ and $\phi$ and $\phi_r$ are characteristic functions. We point out how elements of this class can serve as limit laws as well as some connections between this class and the class of self-decomposable characteristic functions.